A quantitative version of James's Compactness Theorem

Abstract

AbstractWe introduce two measures of weak non-compactness JaE and Ja that quantify, via distances, the idea of boundary that lies behind James's Compactness Theorem. These measures tell us, for a bounded subset C of a Banach space E and for given x* ∈ E*, how far from E or C one needs to go to find x** ∈ $\overline{C}^{w^*}$ ⊂ E** with x**(x*) = sup x*(C). A quantitative version of James's Compactness Theorem is proved using JaE and Ja, and in particular it yields the following result. Let C be a closed convex bounded subset of a Banach space E and r > 0. If there is an element$x_0^{**}$in$\overline{C}^{w^*}$whose distance to C is greater than r, then there is x* ∈ E* such that each x** ∈$\overline{C}^{w^*}$at which sup x*(C) is attained has distance to E greater than ½r. We indeed establish that JaE and Ja are equivalent to other measures of weak non-compactness studied in the literature. We also collect particular cases and examples showing when the inequalities between the different measures of weak non-compactness can be equalities and when the inequalities are sharp.